I'm sure this is asked a lot but....what are my chances of getting into Oxford?

Hi, I'm getting slightly disheartened by my minimal chances of getting into Oxford.

I have mostly As at GCSE (with an A in Mathematics and an A* in History) my only C grade was in Art. I got two As, a B and one B (pending for remark) at AS and have been predicted AAA (History, Government and Politics and Sociology) at A2. I have applied for PPE at Oxford. AAA is the course requirement and according to the prospectus useful (but not required) backgrounds are Maths or History.

The reason I think I'm unlikely to even get an interview is that 1. I did not get 10 A*s at GCSE and 2. I have not (yet) done AS Maths. I could do AS Maths and have been looking into it and practicing but I'd really rather not. That is not to say that I despise Maths, there are aspects of Maths that I really like such as game theory (particularly its relevance to economics and politics) and I can get my head round basic differentiation and derivatives but there's little evidence of that in my personal statement.

That said, I was very happy with how my personal statement turned out (I could share it privately if people would like to look at it to get an idea) but I'm worried that my 'average' GCSEs, especially with a lack of AS Maths, might make Oxford not bother looking at my PS and reject me out of hand because, while I meet the formal requirements, unofficial recommendations I have not met. I think that if I do get an interview I could increase my chances but less than half of PPE applicants do.

The only solution I can think of, if there is one, is to do well enough in the TSA admissions test to get considered for an interview. So I desperately need help with how to pass the TSA with a good mark. Do I yet have a chance or have I thrown away one of my choices biting off more than I can chew?

Any help is incredibly appreciated, thanks